The degenerate Heisenberg category and its Grothendieck ring

Abstract

The degenerate Heisenberg category $\mathcal{H}eis_k$ is a strict monoidal category which was originally introduced in the special case $k=-1$ by Khovanov in 2010. Khovanov conjectured that the Grothendieck ring of the additive Karoubi envelope of his category is isomorphic to a certain $\mathbb{Z}$-form for the universal enveloping algebra of the infinite-dimensional Heisenberg Lie algebra specialized at central charge $-1$. We prove this conjecture and extend it to arbitrary central charge $k \in \mathbb{Z}$. We also explain how to categorify the comultiplication (generically).